\newproblem{lay:1_6_12}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.6.12}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
  Find the general flow pattern of the network shown in the figure. Assuming that the flows are all nonnegative, what
	is the smallest possible value for $x_4$?
	\begin{center}
		\includegraphics[scale=0.5]{Tema2/lay_1_6_12.jpg}
	\end{center}
}{
   % Solution
	To analyze this network we note that at each node the inputs must be equal to its outputs. Consequently:
	\begin{center}
		\begin{tabular}{rc}
		   A & $x_1+x_4=x_2$ \\
			 B & $x_2=x_3+100$ \\
			 C & $x_3+80=x_4$
		\end{tabular}
	\end{center}
	The augmented matrix of this equation system is
	\begin{center}
		$\left(\begin{array}{rrrr|r}
		   1 & -1 &  0 &  1 & 0 \\
			 0 &  1 & -1 &  0 & 100 \\
			 0 &  0 &  1 & -1 & -80 \\
		\end{array}\right)\sim
		\left(\begin{array}{rrrr|r}
		   1 &  0 &  0 &  0 & 20 \\
			 0 &  1 &  0 & -1 & 20 \\
			 0 &  0 &  1 & -1 & -80 \\
		\end{array}\right)$
	\end{center}
	The general solution of this equation system is
	\begin{center}
		$(x_1,x_2,x_3,x_4)=(20,20+x_4,-80+x_4,x_4)$
	\end{center}
	If all flows must be nonnegative, $x_4$ must be at least 80, because otherwise, $x_3$ would be negative.
}
\useproblem{lay:1_6_12}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
